Entropy Power Inequality in Fermionic Quantum Computation
aa r X i v : . [ m a t h - ph ] A ug Entropy Power Inequality in Fermionic QuantumComputation
N. J. B. Aza D. A. Barbosa T.August 14, 2020
Abstract
We study quantum computation relations on unital finite–dimensional
CAR C ∗ –algebras. Weprove an entropy power inequality (EPI) in a fermionic setting, which presumably will permitunderstanding the capacities in fermionic linear optics. Similar relations to the bosonic case areshown, and alternative proofs of known facts are given. Clifford algebras and the Grassmannrepresentation can thus be used to obtain mathematical results regarding coherent fermion states. Keywords:
Quantum Information Theory, Fermionic Gaussian States, Strongly Continuous Semi-groups, Entropy Power.
AMS Subject Classification:
Contents
CAR C ∗ –algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.1 Self–dual CAR algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.2 Grassmann Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.3 Clifford Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 States on
CAR C ∗ –algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 A Open Systems, Quantum Channels and C ∗ –algebras 26 Notation 1.
A norm on the generic vector space X is denoted by k · k X and the identity map of X by X . Thespace of all bounded linear operators on ( X , k · k X ) is denoted by B ( X ) . The unit element of anyalgebra X is always denoted by , provided it exists of course. The set of linear functionals and stateswill be denoted by X ∗ and E X respectively, while its tracial state will be written as tr X . The scalarproduct of any Hilbert space X is denoted by h· , ·i X and Tr X represents the usual trace on B ( X ) . ␅ Introduction
Information Theory is one of the paradigmatic examples in the interphase between physics and math-ematics. Several important results concerning strict aspects in Information Theory have caught theeye of mathematicians. This was revealed when
Shannon founded the
Classical Information Theory (CIT) [Sha01]; among his proposals it was established that the measure of information contained ina physical system can be studied via the probability theory framework. In particular, the measure ofinformation contained in a random variable can be described by
Shannon’s entropy . Some results ofinterest in CIT are the
Classical Young’s Inequality , de Bruijin identity and the Stam inequality . Asstressed by Blachman [Bla65], all these serve as springboards to prove the
Convolution Inequality forEntropy Powers , which from a physical point of view, is useful to determine information capacitiesof broadcast channels [KS14], for instance. See below, Expression (1) for a concrete overview. Fromthe mathematical point of view, such
Entropy Power Inequalities (EPI) are interesting due to theirconnection with geometrical quantities such as the
Brunn–Minkowski inequalities , which bound thevolume of the set–sum of two compact convex sets in R d , with d ∈ N . For further details about theimportance of EPI in physics and mathematics see [DMG14] and [Gar02], respectively.In the quantum setting, one replaces probability densities by density matrices , defined on someHilbert space H : ρ H . = { ρ ∈ B ( H ) : ρ > with Tr H ( ρ ) = 1 } . These density matrices areuseful to describe a non–commutative probability measure space in quantum systems. Among therecent analogs to classical probabilities and their quantum counterpart the Fokker–Planck equation has been stated. Namely, in [CM14], Carlen and Maas used the
Clifford C ∗ –algebras as a probabilityspace, and developed a differential calculus to get a fermionic Fokker–Plank equation. They gave anensemble of parallel results between the quantum and classical cases.In this work we prove an EPI for fermion systems, and we provide pivotal identities such as deBruijin’s identity and the Stam inequality for this non–commutative framework. As stressed in thePh.D. Thesis [Aza17], in a fermionic setting, geometric inequalities and their relation with quantuminformation are unknown. We focus on a quantum information framework in the scope of fermionicquantum information, namely, fermionic linear optics (FLO). The latter refers to free –fermion sys-tems under external potentials, which is reminiscent of simple conduction electron problems as atthe
Anderson model [And58, KLM07, KM08, KM14, BPH14]. Physically, FLO is a limited form ofquantum computation that can efficiently simulate classical computers and their study might help atthe understanding of quantum channels [Bra05, BK12]. As a one of the main contributions of thispaper is to construct an unambiguous mathematical structure to study at FQI, which, in turn, is pivotalto prove an EPI at the fermionic setting.In order to state the relevance of the problem, let (Ω , Σ , m ) be a probability measure space. Forrandom variables X and Y in (Ω , Σ , m ) , one clasically study information quanties such as the “Shan-non (differential) entropy” H ( X ) and the “convolution” between X and Y , u X + Y . The latter aredefined by [Sha01] H ( X ) . = − Z u X ( x ) ln u X ( x ) m (d x ) u X + Y ( z ) . = Z u Y ( z − x ) u X ( x ) m (d x ) , with z ∈ Ω and u X the probability density associated to X . Note that u X + Y can be recognized as theprobability density of the output X + Y . Then, if X . = { X j } ≤ j ≤ d and Y . = { Y j } ≤ j ≤ d are two setsof random variables with their joint respective probability measures on R d ∈ N , the Shannon entropy ofthe combination of the variables X and Y satisfies the entropy power inequality for classical systems e d H ( X + Y ) ≥ e d H ( X ) + e d H ( Y ) . In [KS14], K¨onig and Smith improved last inequality in the sense that for λ ∈ C ≡ [0 , they proventhe following two concave inequalities:(1) e d H ( X ⊞ λ Y ) ≥ λ X e d H ( X ) + λ Y e d H ( Y ) , H ( X ⊞ λ Y ) ≥ λ X H ( X ) + λ Y H ( Y ) , ⊞ λ refers to the addition rule ⊞ λ : R d × R d → R over the probability density spaceobeying u X ⊞ λ Y . = u √ λ X X + √ λ Y Y with λ X . = λ and λ Y . = 1 − λ , the weights of the sets of randomvariables X and Y , respectively. In order to prove the inequalities (1), they use the heat or diffusion(or Gaussian) equation, which in the current context is described as follows: for ( X, t ) ∈ R d +1 with d ∈ N , the heat or diffusion (or Gaussian) equation describing a scalar field u ( X, t ) : R d × R + → R has the form of the Cauchy problem (2) ∂u ( X, t ) ∂t = ∆ u ( X, t ) , u ( X ) . = u ( X, , where u ( X ) is the initial data and ∆ . = d P i =1 ∂ ∂x i is the Laplacian . By [EBN +
06, Chapter 2, Sect.2] we know that for any p ∈ [1 , ∞ ) the diffusion semigroup , { P t } t ∈ R + ∈ L p ( R d ) , associated to thelatter equation verifies for u ∈ L p ( R d ) that P t u ( X ) = 1(4 πt ) d Z R d e − k X − Y k t u ( Y )d Y. Then, one can prove that if P = 1 , the family of semigroups P . = { P t } t ∈ R + ∈ L p ( R d ) is stronglycontinuous and even more, the solution of (2) is provided by u ( X ) and P via u ( X, t ) = P t u ( X ) .See Definition 11 at Appendix A for a general formulation of semigroups. Note that in [KS14],is assumed that the time dependency of the random set X is X t = X + √ tZ , for t ∈ R +0 and Z . = { Z j } ≤ j ≤ d is a set of random variables with standard normal distribution, N (0 , .Quantum bosonic versions of (1)–(2) were given in [KS14] and [DMG14], and hence the mathemat-ical framework of CCR algebras was required. They consider bosonic systems interacting with theenvironment such that dissipative processes can occur. The latter means that given some initial den-sity matrix ρ ≡ ρ on a well–defined subset of a CCR C ∗ –algebra W ( ρ satisfies tr W ( ρ ) = 1 and itis positive), the density matrix ρ t evolves over time via the Markovian master equation (see (62) and(80))(3) dd t ρ t = L ρ t , ρ t . = e t L ρ, for any t ∈ R +0 . Here, L is the so–called Liouvillean of the system, which is the infinitesimal(unbounded) generator of the strongly continuous semigroup P . = { e t L } t ∈ R +0 . By assuming that the CCR algebra has a unit operator , explicitly, for any A ∈ D ( L ) (domain of L ), L is given by L A = − X i ∈ I ([ Q i , [ Q i , A ]] + [ P i , [ P i , A ]]) , where I is a finite index set, and { Q i } i ∈ I , { P i } i ∈ I are two families of operators satisfying the CCR relations [ Q i , P j ] = i δ i,j , [ Q i , Q j ] = 0 = [ P i , P j ] , for i, j ∈ I . As is usual, Q i and P i are the position and momentum operators at the i – mode . Here, [ A, B ] . = AB − BA denotes the commutator between A and B .Instead of using for λ ∈ C ≡ [0 , the addition rule ⊞ λ described above, they consider the beamsplitter quantum channel M U λ defined by:(4) ρ C ≡ M U λ ( ρ A ⊗ ρ B ) . = tr B ( U ∗ λ ( ρ A ⊗ ρ B ) U λ ) , More generally the heat equation is the simplest example of parabolic equation [Bre10]. The name refers to
Canonical Commutation Relations . ρ A and ρ B are the density matrices of two different but similar bosonic systems, e.g., thesystem A is described by the CCR C ∗ –algebra W A generated by the family of (unbounded) opera-tors { a i } i ∈ I while the family of (unbounded) operators { b i } i ∈ I generates the CCR C ∗ –algebra W B ,which describes the bosonic system B . In fact, they both satisfy the CCR ( C ∗ –algebra) relations [ a i , a ∗ j ] = δ i,j , [ b i , b ∗ j ] = δ i,j , with i, j ∈ I . Here, as is usual, for
A, B ∈ W A , B , [ A, B ] . = AB − BA ∈ W A , B denotes the commutator between A and B , which by simplicity we assume that is closed on W A , B . Moreover, for any a i ∈ W A and b j ∈ W B we impose [ a i , b j ] = 0 . In (4), for λ ∈ C , U λ is a unitary operator implementing the beam–splitter ∗ –automorphism on the tensor product algebra W A ⊗ W B so that U λ U ∗ λ = , and tr B ( · ) meansthat one of the output signals (in this case of the system B ) is discarded in order to get the output mixed density matrix ρ C . Additionally, under the quantum channel M U λ the mixing of the families { a i } i ∈ I and { b i } i ∈ I provides a new family of (unbounded) operators { c i } i ∈ I : c i = q λ A a i + q λ B b i , i ∈ I , which uphold the CCR relations, i.e., [ c i , c j ] = δ i,j , with λ A . = λ, λ B . = 1 − λ . For more detailsabout channels in the C ∗ –algebra context, see Appendix A.Then the modified bosonic version of the entropy power inequality as given in (1) is the followingconcave inequality: E ( C ) ≥ λ A E ( A ) + λ B E ( B ) , where for D ∈ { A , B , C } , ρ D is a density matrix, E ( D ) . = e s ( ρ D ) is the quantum entropy power with s ( ρ D ) . = | I | S ( ρ D ) the density entropy of ρ D such that S ( ρ D ) . = − tr W ( ρ D ln ρ D ) , denotes the von Neumann entropy.In contrast to the mentioned results, the current work is focused in its fermionic version: Ourcontributions regard the study of fermionic systems in a quantum computation framework. From thephysical point of view, this is of interest because as already mentioned FLO is classically simulable.Mathematically, one must consider a non–commutative framework and hence it is interesting to studythe existence of the inequality (1) for the underlying setting.The standard mathematical formalism when one deals with fermion systems are CAR C ∗ –algebras.If one consider a unital finite CAR C ∗ –algebra A ≡ ( A , ∗ , + , · , k · k A ) of size N with N ∈ N ,then this is isomorphic to the C ∗ –algebra of the square complex matrices Mat(2 N , C ) [BR03b]. Fora finite Hilbert space h , two very well–studied CAR C ∗ –algebras are that given by (i) the usual CAR algebra U ≡ U ( h ) generated by the identity and the elements { a ( ϕ ) } ϕ ∈ h such that satisfy the CAR (5) a ( ϕ ) a ( ϕ ) ∗ + a ( ϕ ) ∗ a ( ϕ ) = h ϕ , ϕ i , ϕ , ϕ ∈ h , and (ii) the Clifford
CAR algebra Q ≡ Q ( h ) , which is generated by and the self–adjoint elements { R ± ( ϕ ) } ϕ ∈ h so that R ( ϕ ) R ( ϕ ) + R ( ϕ ) R ( ϕ ) = 2 h ϕ , ϕ i δ + , − , ϕ , ϕ ∈ h , where denotes either R + or R − . Naturally, U and Q are isomorphic [CL93]. Note that, viewcan algebras, U and Q are isomorphic to the Grassmann algebra ∧ ∗ h because they have exactly thesame dimension: dim ∧ ∗ h = 2 dim h = dim U = dim Q . Moreover, by using Definition 5 below,we can endow to Grassmann algebras of a well–defined norm (see expression 29) such that the C ∗ –algebras dim ∧ ∗ h = 2 dim h = dim U = dim Q are equivalent. The latter gain relevance for fermionic oherent states because we are able to prove transparent properties associated to the displacement fermion operator, which are reminiscent to the bosonic version applied on quantum optics [CG99].See Definition 9 and Lemma 2 below.Our main result is Theorem 1, and the set of Lemmata 2, 3, 4 and 5 are pivotal. Theorem 1 refers tothe fermionic entropy power inequality for a beam–splitter quantum channel operation. Its extensionto amplifiers quantum channels follows directly from the methods used in [DMG14]. We omit themathematical details for the sake of simplicity. Additionally, we prove Lemmata 3 and 4 even for infi-nite dimensional CAR C ∗ –algebras, which differ in their proofs of their parallel bosonic counterpartgiven in [HKV17, DPR17].To conclude, this paper is organized as follows:• In Section 2, we present the mathematical framework of CAR C ∗ –algebras. As a first stepwe present the self–dual CAR C ∗ –algebras introduced by Araki [Ara68, Ara71], which wereelegantly raised in the study of non –interacting but non –gauge fermion systems. Secondly,we provide the well–known Grassmann algebras G , as well as the Berezin integrals and wedefine a “circle” product such that it converts the self–dual CAR algebras and the Grassmannalgebras to be equivalent. After that, the Clifford C ∗ –algebras, isomorphic to U and G , areintroduced. Then, a non –commutative calculus is shown, which will provide us the differentialequation determining the evolution of fermionic open systems. At the end of the section generalproperties of the set of states are exposed.• Section 3 provides the statement of the main result, Theorem 1, and some main definitionsconcerning the entropy for the current work are defined. To be precise, we conveniently use theClifford C ∗ –algebra to define information quantities such as the quantum Fisher informationand the entropy variation rate.• Section 4 devotes to all technical proofs. Fermionic versions of well–known bosonic versionsare proven. Additionally, as an application of our results, in the fermionic coherent statescontext, is the derivation of a natural mathematical framework only requiring the circle productpresented in Section 2.• We finally include Appendix A, stating a C ∗ –algebra mathematical framework on the study ofopen quantum systems. This will permit to the non–experts at quantum information theory andopen quantum systems understand their mathematical motivations. CAR C ∗ –algebras CAR algebras
From now on and through of all the paper, let H be a finite–dimensional (complex) Hilbert spacewith even dimension dim H ∈ N , and let A be an antiunitary involution on H such that A = H and h A ϕ , A ϕ i H = h ϕ , ϕ i H , ϕ , ϕ ∈ H . H endowed with A , denoted ( H , A ) is named a self–dual Hilbert space and yields self–dual CAR algebra , U ≡ (sCAR( H , A ) , + , · , ∗ , k · k sCAR( H , A ) ) , which is nothing but a C ∗ –algebra generatedby a unit and a family { B( ϕ ) } ϕ ∈ H of elements satisfying: B ( ϕ ) ∗ is (complex) linear, B( ϕ ) ∗ = ( A ( ϕ )) for any ϕ ∈ H and the family { B( ϕ ) } ϕ ∈ H satisfies the Canonical Anti–CommutationRelations ( CAR ): For any ϕ , ϕ ∈ H ,(6) B( ϕ )B( ϕ ) ∗ + B( ϕ ) ∗ B( ϕ ) = h ϕ , ϕ i H . Note that for any ϕ ∈ H , k B( ϕ ) k U ≤ k ϕ k H with k · k U ≡ k · k sCAR( H , A ) . Additionally, U is isomorphic to the C ∗ –algebra ⊗ dim H / Mat(2 , C ) , where for N ∈ N , Mat( N, C ) denotes thecomplex matrices of size N × N , see (18) and [BR03b].For any self–dual Hilbert space ( H , A ) we introduce: Definition 1 (Basis projections).
A basis projection associated with ( H , A ) is an orthogonal projec-tion P ∈ B ( H ) satisfying A P A = P ⊥ . = H − P . We denote by h P the range ran P of the basisprojection P . The set of all basis projections on ( H , A ) it will denoted by p ( H , A ) . ␈ For any P ∈ p ( H , A ) , we can identify H with(7) H ≡ h P ⊕ h ∗ P and(8) B ( ϕ ) ≡ B P ( ϕ ) . = B ( P ϕ ) + B (cid:16) A P ⊥ ϕ (cid:17) ∗ . Therefore, there is a natural isomorphism of C ∗ –algebras from U to the CAR algebra
CAR( h P ) generated by the unit and { B P ( ϕ ) } ϕ ∈ h P . See Expression (5). In other words, a basis projection P can be used to fix so–called annihilation and creations operators.For any unitary operator U ∈ B ( H ) such that U A = A U , the family of elements B( U ϕ ) ϕ ∈ H ,together with the unit , generates U . In the latter case, U is named a Bogoliubov transformation ,and the unique ∗ –automorphism χ U such that(9) χ U (B( ϕ )) = B( U ϕ ) , ϕ ∈ H , is called in this case a Bogoliubov ∗ –automorphism . Note that a Bogoliubov transformation U ∈ B ( H ) always satisfies: det ( U ) = ± . If det ( U ) = 1 , we say that U has positive orientation.Otherwise U is said to have negative orientation. These properties are also called even and odd.Considering the Bogoliubov ∗ –automorphism (9) with U = − H , an element A ∈ U , satisfying(10) χ − H ( A ) = A is called even , − A is called odd . The subspace of even elements U + is a sub– C ∗ –algebra of U .In order to study non –interacting fermion systems, as is the case of the (reduced) BCS modelat condensed matter physics, or Gaussian states at fermionic quantum computation, it is useful tointroduce for H ∈ B ( H ) its bilinear element by h B , H B i . = X i,j ∈ I D ψ i , Hψ j E H B (cid:16) ψ j (cid:17) B ( ψ i ) ∗ , (11)where { ψ i } i ∈ I is an orthonormal basis of H . Note that h B , H B i is uniquely defined in the sense that does not depend on the particular choice of the orthonormal basis, but does depend on the choice ofgenerators { B( ϕ ) } ϕ ∈ H of the self–dual CAR algebra U . Moreover, h B , H B i ∗ = h B , H ∗ B i for all H ∈ B ( H ) . The analysis of bilinear elements can be restricted to self–dual operators:6 efinition 2 (Self–dual operators). A self–dual operator on ( H , A ) is an operator H ∈ B ( H ) sat-isfying the equality H ∗ = − A H A . If, additionally, H is self–adjoint, then we say that it is a self–dualHamiltonian on ( H , A ) . ␈ A basis projection P (Definition 1) (block–) “diagonalizes” the self–dual operator H ∈ B ( H ) whenever(12) H = 12 (cid:16) P H P P − P ⊥ A H ∗ P A P ⊥ (cid:17) , with H P . = 2 P HP ∈ B ( h P ) . In this situation, we also say that the basis projection P diagonalizes h B , H B i . On the other hand,given some self–dual Hamiltonian H ∈ B ( H ) , and basis projection P ∈ p ( H , A ) with range h P ,one can define a self–adjoint operator H P = H ∗ P ∈ B ( h P ) and the antilinear operator G ∗ P = − G P on h P as(13) H P . = 2 P HP and G P . = 2 P H A P. With this notation h P is the so–called one–particle Hilbert space while H P and G P are the gauge in-variant and not gauge invariant one–particle Hamiltonians respectively. In fact, for elements { ϕ i } i ∈ J ∈ h P , in the CAR C ∗ –algebra setting (with generators and { a ( ϕ i ) } i ∈ J , see Expression 5), one canwrite any quadratic fermionic Hamiltonian as linear combinations of gauge–invariant elements a ( ϕ i ) a ( ϕ j ) ∗ ,for all i, j ∈ J , and linear combinations of non–gauge–invariant elements a ( ϕ i ) a ( ϕ j ) and a ( ϕ i ) ∗ a ( ϕ j ) ∗ ,for all i, j ∈ J . Then, from (11) one note that any quadratic fermionic Hamiltonian can be recognizedas dΓ( H P ) + dΥ( G P ) = −h B , [ κ ( H P ) + ˜ κ ( G P )] B i + 12 Tr h P ( H P ) , with κ ( H P ) . = 12 ( P H P P − A P H P P A ) ∈ B ( H ) , ˜ κ ( G P ) . = 12 ( P G P P A − A P G P P ) ∈ B ( H ) , and dΥ( G P ) = − h B , ˜ κ ( G P ) B i . Additionally, the CAR C ∗ –algebra CAR( h P ) and the self–dual CAR C ∗ –algebra sCAR( H , A ) are the same C ∗ –algebra, by defining B ( ϕ ) . = a ( ϕ ) + a ( ϕ ) ∗ , ϕ = ( ϕ , ϕ ∗ ) ∈ h P ⊕ h ∗ P . Consider the self–dual Hilbert space ( H , A ) . Grassmann algebras, also called exterior algebras aredefined as follows: For every n ∈ N and ϕ , . . . , ϕ n ∈ H , we define the completely antisymmetric n -linear form ϕ ∧ · · · ∧ ϕ n from H n to C by ϕ ∧ · · · ∧ ϕ n ( ψ , . . . , ψ n ) . = det (cid:16) ( ϕ ∗ k ( ψ l )) nk,l =1 (cid:17) = det (cid:16) ( h ϕ k , ψ l i H ) nk,l =1 (cid:17) , ψ , . . . , ψ n ∈ H . Then, using the definitions ∧ ∗ H . = C and, for n ∈ N ,(14) ∧ ∗ n H . = lin { ϕ ∧ · · · ∧ ϕ n : ϕ , . . . , ϕ n ∈ H ∗ ≡ H } , we denote by(15) ∧ ∗ H . = ∞ M n =0 ∧ ∗ n H ≡ ( ∧ ∗ H , + , ∧ ) ( H , A ) . Here, the exterior product is defined, for any n, m ∈ N , ξ ∈ ∧ ∗ n H and ζ ∈ ∧ ∗ m H , by ξ ∧ ζ (cid:16) ψ , . . . , ψ n + m (cid:17) . = 1 n ! m ! X π ∈ S n + m ( − π ξ (cid:16) ψ π (1) , . . . , ψ π ( n ) (cid:17) ζ (cid:16) ψ π ( n +1) , . . . , ψ π ( n + m ) (cid:17) , where S N is the set of all permutations of N ∈ N elements and ψ , . . . , ψ n + m ∈ H . Obviously, forany n ∈ N , n ≥ , and ϕ , . . . , ϕ n ∈ H ,(16) ϕ ∧ ϕ = − ϕ ∧ ϕ and ϕ ∧ ( ϕ ∧ · · · ∧ ϕ n ) = ϕ ∧ · · · ∧ ϕ n . In the sequel, when there is no risk of ambiguity, we use(17) ϕ ∧ · · · ∧ ϕ n ≡ ϕ · · · ϕ n , ϕ , . . . , ϕ n ∈ H . The unit of the Grassmann algebra ∧ ∗ H is denoted by . = 1 ∈ ∧ ∗ H ⊆ ∧ ∗ H and [ ξ ] n stands for the n -degree component of any element ξ of ∧ ∗ n H , with n ∈ N . Note also that H ≡ H ∗ . = ∧ ∗ H . The subspace of ∧ ∗ H generated by monomials ϕ · · · ϕ n of even order n ∈ N forms a commutativesubalgebra, the even subalgebra of ∧ ∗ H , which is denoted by ∧ ∗ + H in the sequel.For any (complex) Hilbert space H and antiunitary involution A on H we can define a self–dualCAR algebra U . The linear spaces U and ∧ ∗ H are isomorphic to each other because they haveexactly the same dimension:(18) dim U = 2 dim H = dim ( ∧ ∗ H ) . However, because of the CAR (6) and the involution A , ( ∧ ∗ H , + , ∧ ) is not isomorphic to a self–dual CAR algebra over H and, following [ABPM20], we introduce the circle product at Definition 5 aswell as an involution in order to make ∧ ∗ H a self–dual CAR algebra.For any ϕ ∈ H , the linear operator δ/δϕ acting on the Grassmann algebra ∧ ∗ H is called Berezinderivative , which is uniquely defined by the conditions(19) δδϕ ˜ ϕ = h ϕ, ˜ ϕ i H and δδϕ ξ ξ = δδϕ ξ ! ∧ ξ + ( − n ξ ∧ δδϕ ξ ! , for any ˜ ϕ ∈ H and element ξ ∈ ∧ ∗ n H of degree n ∈ N , and all ξ ∈ ∧ ∗ H .For each k ∈ N , H ( k ) denotes a copy of the Hilbert space H and the corresponding copy of ξ ∈ H is written as ξ ( k ) . For any K ⊂ { , . . . , N } with N ∈ N , we identify ∧ ∗ ( ⊕ k ∈ K H ( k ) ) with theGrassmann subalgebra of ∧ ∗ ( ⊕ Nk =0 H ( k ) ) generated by the union [ k ∈ K n ϕ ( k ) : ϕ ∈ H o . We meanwhile identify ∧ ∗ H (0) with the Grassmann algebra ∧ ∗ H , i.e.,(20) ∧ ∗ H (0) ≡ ∧ ∗ H . Taking into account this, we define: 8 efinition 3 (Berezin integral).
Let N ∈ N and consider a basis projection P (Definition 1) with { ψ i } i ∈ J being any orthonormal basis of its range h P . For all k ∈ { , . . . , N } , we define the linearmap Z P d (cid:16) H ( k ) (cid:17) : ∧ ∗ (cid:16) ⊕ Nq =0 H ( q ) (cid:17) → ∧ ∗ (cid:16) ⊕ q ∈{ ,...,N }\{ k } H ( q ) (cid:17) by Z P d (cid:16) H ( k ) (cid:17) . = Y i ∈ J δδψ ( k ) i δδ (( A ψ i ) ( k ) ) ! . ␈ For N = 0 , the Berezin integral defines a linear form from ∧ ∗ H (0) ≡ ∧ ∗ H to C ≡ C . One canshow that for any basis projection P diagonalizing a self–dual operator H ∈ B ( H ) , we have(21) Z P d ( H ) e h H ,H H i = det ( H P ) , where for an orthonormal basis { ψ i } i ∈ I of H , the bilinear element h H , H H i on the Grassmannalgebra ∧ ∗ H is uniquely given by(22) h H , H H i . = X i,j ∈ I D ψ i , Hψ j E H (cid:16) A ψ j (cid:17) ∧ ψ i , c.f. (11). In particular, det ( H P ) only depends on H and the orientation of P , which was definedaround expression (9). Gaussian Berezin integrals are then defined as follows: Definition 4 (Gaussian Berezin integrals).
For C ∈ B ( H ) an invertible self–dual operator, the Gaus-sian Berezin integral with covariance C ∈ B ( H ) is the linear map R d µ C ( H ) from ∧ ∗ H to C defined by Z d µ C ( H ) ξ . = det (cid:18) C P (cid:19) Z P d ( H ) e h H ,C − H i ∧ ξ, ξ ∈ ∧ ∗ H , where P ∈ p ( H , A ) is any basis projection diagonalizing C (see (12)). ␈ It can be proven the following [ABPM20]:P
ROPOSITION
AUSSIAN B EREZIN INTEGRALS AS P FAFFIANS ).Let C ∈ B ( H ) be any invertibleself–dual operator. Then, R d µ C ( H ) = while, for all N ∈ N and ϕ , . . . , ϕ N ∈ H , Z d µ C ( H ) ϕ · · · ϕ N = 0 and Z d µ C ( H ) ϕ · · · ϕ N = Pf [ h A ϕ k , Cϕ l i H ] Nk,l =1 . ␊ Take a basis projection P ∈ p ( H , A ) with range h P . For all i, j, k, l ∈ N :(23) κ ( k,l )( i,j ) : ∧ ∗ ( h ( i ) P ⊕ h ∗ ( j ) P ) → ∧ ∗ ( h ( k ) P ⊕ h ∗ ( l ) P ) is the unique isomorphism of linear spaces such that κ ( k,l )( i,j ) ( z ) = z for z ∈ C and, for any m, n ∈ N so that m + n ≥ , and all ϕ , . . . , ϕ m + n ∈ h P ,(24) κ ( k,l )( i,j ) (cid:16) ( A ϕ ) ( i ) · · · ( A ϕ m ) ( i ) ϕ ( j ) m +1 · · · ϕ ( j ) m + n (cid:17) = ( A ϕ ) ( k ) · · · ( A ϕ m ) ( k ) ϕ ( l ) m +1 · · · ϕ ( l ) m + n with ϕ ∧ ϕ ≡ ϕ ϕ .We can equip any Grassmann algebra with a C ∗ –algebra structure. In order to proceed we take a basisprojection P ∈ p ( H , A ) in such a way that we introduce the circle product ◦ P as follows:9 efinition 5 (Circle products with respect to basis projections). Fix P ∈ p ( H , A ) with range h P andrecall (7), that is, H ≡ h P ⊕ h ∗ P . For any ξ , ξ ∈ ∧ ∗ H , we define their circle product by ξ ◦ P ξ . = ( − dim H Z P d (cid:16) H (1) (cid:17) κ (0 , , ( ξ ) κ (1 , , ( ξ )e −h h (0) P , h (0) P i e h h (0) P , h (1) P i e −h h (1) P , h (1) P i e h h (1) P , h (0) P i . ␈ The space ( ∧ ∗ H , +) endowed with the circle product ◦ P is an (associative and distributive) algebra,like ( ∧ ∗ H , + , ∧ ) , for any P ∈ p ( H , A ) . Among other properties of ◦ P note that this satisfy theCanonical Anti-commutation Relations ( CAR ):(25) ϕ ∗ ◦ P ϕ + ϕ ◦ P ϕ ∗ = h ϕ , ϕ i H , ϕ , ϕ ∈ H . Furthermore, for any P ∈ p ( H , A ) , and ϕ , ϕ ∈ h P ,(26) ( A ϕ ) ◦ P ϕ = ( A ϕ ) ∧ ϕ . We can endow ∧ ∗ H with an involution, which turns ( ∧ ∗ H , + , ◦ P ) into a ∗ –algebra. Namely, onedefine the involution to satisfy for any P ∈ p ( H , A ) that ∗ = and ( ϕ ◦ P ϕ ) ∗ = ( A ϕ ) ◦ P ( A ϕ ) , n ∈ N , ϕ , ϕ ∈ H . Hence, ( ∧ ∗ H , + , ∧ ) equipped with the involution ∗ is a ∗ –algebra, i.e.,(27) ( ξ ∧ ξ ) ∗ = ξ ∗ ∧ ξ ∗ , ξ , ξ ∈ ∧ ∗ H . Thus for a self–dual Hilbert space ( H , A ) and P ∈ p ( H , A ) , ( ∧ ∗ H , + , ◦ P , ∗ ) is a ∗ –algebra gener-ated by and the family { ϕ ∗ } ϕ ∈ H of elements satisfying the same properties of the self–dual CAR algebras (see Expression (6)), namely, (25). Additionally, there is a canonical ∗ –isomorphism betweena self–dual CAR algebra constructed from ( H , A ) and ∧ ∗ H : Definition 6 (Canonical isomorphism of ∗ –algebra). For P ∈ p ( H , A ) , we define the canonical iso-morphism κ P : ( U , + , · , ∗ ) → ( ∧ ∗ H , + , ◦ P , ∗ ) via the conditions κ P ( z ) = z and κ P (B( ϕ )) = ϕ ∗ for all ϕ ∈ H . ␈ Therefore note that bilinear elements of self–dual
CAR algebra, see Equation (11), are mapped via κ P (up to some constant) to bilinear elements of Grassmann algebra, as stated in Definition 9. In fact,one can proof that for any P ∈ p ( H , A ) ,(28) κ P ( h B , H B i ) = h H , H H i + Tr H (cid:16) P ⊥ HP ⊥ (cid:17) , H ∈ B ( H ) . For P ∈ p ( H , A ) , we endow ( ∧ ∗ H , + , ◦ P , ∗ ) with the norm(29) k ξ k ∧ ∗ H . = (cid:13)(cid:13)(cid:13) κ − P ( ξ ) (cid:13)(cid:13)(cid:13) U , ξ ∈ ∧ ∗ H , in order to do it a self–dual CAR ( C ∗ –) algebra. In this case, κ P is an isometry. In the sequel wewill write G P ≡ ( ∧ ∗ H , + , ◦ P , ∗ , k·k ∧ ∗ H ) , to lighten notation. Similarly, for the commutative evensubalgebra ∧ ∗ + H associated to ∧ ∗ H (see (16) and comments around it), from now on, G + P will be acommutative C ∗ –algebra. 10 .1.3 Clifford Algebras Clifford C ∗ –algebras are presented in the following way: For any basis projection P ∈ p ( H , A ) , andany orthonormal basis { ψ j } j ∈ J of range h P , we define the self–adjoint elements Q j ≡ Q ( ψ j ) . = B( ψ j ) ∗ + B( ψ j ) and P j ≡ P ( ψ j ) . = i(B( ψ j ) ∗ − B( ψ j )) , for any j ∈ J . The family of elements { Q j } j ∈ J (resp. { P j } j ∈ J ) is known as the configurationoperators (resp. conjugate momenta operators ) [CL93]. At the fermionic information context, | J | =dim H / is the number of fermionic modes of the physical system, while the operators { Q j } j ∈ J and { P j } j ∈ J are known as the Majorana fermion operators. Let(30) J . = J × { + , −} . We denote by R j, + = Q j and R j, − = P j , for any j ∈ J . Note that the unit and the family ofself–adjoint elements { R j } j ∈ J generate the Clifford algebra Q ≡ ( Q , + , · , ∗ , k · k Q ) of size dim Q =2 dim H , satisfying the CAR :(31) R i R j + R j R i = 2 δ i , j with δ i , j . = δ i,j δ s,t . for s, t ∈ { + , −} . Note that C ∗ –algebras U (self–dual CAR algebra) and Q have exactly the samedimension, and then are isomorphic, thus by (18) and Definition 6, Q is also isomorphic to the C ∗ –algebra G . Furhermore, similar to the algebras U and ∧ ∗ H cases, for any H ∈ B ( H ) one can uniquely introduce the bilinear element h R , H R i on Q by(32) h R , H R i . = X i , j ∈ J h ψ i , Hψ j i H R ( ψ j ) R ( ψ i ) , c.f. (11) and (22), where { ψ i } i ∈ J is an orthonormal basis of H , with J given by (30).We can endow Q with the Hilbert–Schmidt inner product h· , ·i H.S. Q given by(33) h A, B i H.S. Q . = tr Q ( A ∗ B ) , A, B ∈ Q , where tr Q is the tracial state on Q . Consider now the Bogoliubov ∗ –automorphism χ − H : U → U given by (10) (and hence χ − H : Q → Q too), for j ∈ J , in order to introduce the skew–derivation ∇ j : Q → Q as ∇ j ( A ) . = 12 (cid:16) R j A − χ − H ( A ) R j (cid:17) , A ∈ Q . Here, by skew we mean that for each
A, B ∈ Q we have and anti–derivation Leibniz’s law property ∇ j ( AB ) = ∇ j ( A ) B + χ − H ( A ) ∇ j ( B ) . Additionally, since χ − H ( R j ) = − R j for any j ∈ J we can consider the inner product h· , ·i H.S. Q , suchthat h∇ j ( A ) , B i H.S. Q = h A, ∇ ∗ j ( B ) i H.S. Q , and D χ − H ( A ) , BR j E H.S. Q = − D A, χ − H ( B ) R j E H.S. Q in order toget ∇ ∗ j ( A ) = 12 (cid:16) R j A + χ − H ( A ) R j (cid:17) , A ∈ Q . Then, for any
A, B ∈ Q , one introduce the fermionic number operator N on Q satisfying,(34) F ( A, B ) . = h A, N B i H.S. Q , See [CM20] for a general study of non–commutative calculus at the framework of C ∗ –algebras. F is the Gross’s Fermionic Dirichlet form F ( A, A ) on Q so that [CM14] F ( A, A ) . = tr Q (( ∇ A ) ∗ · ∇ A ) = X j ∈ J tr Q (( ∇ j ( A )) ∗ · ∇ j ( A )) . Comparing the two latter equations, one note that it is possible to write F ( A, A ) = X j ∈ J tr Q (cid:18)(cid:18) (cid:16) R j A − χ − H ( A ) R j (cid:17)(cid:19) ∗ · (cid:18) (cid:16) R j A − χ − H ( A ) R j (cid:17)(cid:19)(cid:19) = 14 X j ∈ J tr Q (cid:16) A ∗ (cid:16) A − R j χ − H ( A ) R j (cid:17) + χ − H ( A ) ∗ (cid:16) χ − H ( A ) − R j AR j (cid:17)(cid:17) , and using that D χ − H ( A ) , A E H.S. Q = D A, χ − H ( A ) E H.S. Q we obtain from (34) that(35) N A = 12 X j ∈ J tr Q (cid:16) A − R j χ − H ( A ) R j (cid:17) , such that we finally introduce the “Fermionic Mehler semigroup” as { P t } t ∈ R +0 . = n e − t N o t ∈ R +0 . Thissemigroup satisfies the differential equation dd t ρ t = − N ρ t , ρ t . = e − t N ρ, for any density matrix ρ ∈ Q + ∩ Q and ρ . = ρ . CAR C ∗ –algebras Take N ∈ N . If M ∈ Mat (2 N, C ) is a complex matrix of size N × N and satisfies M k,l = − M l,k is said to be “skew–symmetric” or “anti–symmetric”. If additionally M is a normal matrix ( M M ∗ = M ∗ M ) then exists a unitary U ∈ Mat(2 N, C ) with U t denoting its transpose such that M = U t Λ U ,where Λ is a block diagonal matrix of size N × N such that it can be decomposed as a direct sumof N skew–symmetric matrices of size × . More precisely(36) Λ . = n M j =1 Λ j ≡ diag { Λ , . . . , Λ N } , where, for j ∈ { , . . . , N } , Λ j is a skew–symmetric matrix with entries { Λ j } = − { Λ j } = λ j ∈ R . Note that for ( H , A ) a self–dual Hilbert space, a self–dual operator C ∈ B ( H ) and N ∈ N , thecomplex matrix defined by(37) C k,l . = h A ϕ k , Cϕ l i H , k, l ∈ { , . . . , N } is skew–symmetric.We again consider the self–dual Hilbert space ( H , A ) , and consider the set of states of U (theself–dual CAR C ∗ –algebra associated to ( H , A ) ), denoted by E U ⊂ U ∗ . An important class ofstates are the so–called quasi–free states, that are defined for all N ∈ N and ϕ , . . . , ϕ N ∈ H as(38) ω (B ( ϕ ) · · · B ( ϕ N )) = 0 , while, for all N ∈ N and ϕ , . . . , ϕ N ∈ H ,(39) ω (B ( ϕ ) · · · B ( ϕ N )) = Pf [ ω ( O k,l (B( ϕ k ) , B( ϕ l )))] Nk,l =1 , See Appendix A for a general discussion of states on the C ∗ –algebra setting. Pf is Pfaffian of the N × N skew–symmetric matrix M ∈ Mat (2 N, C ) defined by(40) Pf [ M k,l ] Nk,l =1 . = 12 N N ! X π ∈ S N ( − π N Y j =1 M π (2 j − ,π (2 j ) and O k,l by O k,l ( A , A ) . = A A for k < l, − A A for k > l, for k = l. Quasi–free states are therefore particular states that are uniquely defined by two-point correlationfunctions, via (38)–(39). In fact, a quasi–free state ω ∈ E U is uniquely defined by its so–called symbol , that is, a positive operator S ω ∈ B ( H ) such that(41) ≤ S ω ≤ H and S ω + A S ω A = H , through the conditions(42) h ϕ , S ω ϕ i H = ω (B( ϕ )B( A ϕ )) , ϕ , ϕ ∈ H . For more details on symbols of quasi–free states, see [Ara71, Lemma 3.2]. Conversely, any self–adjoint operator satisfying (41) uniquely defines a quasi–free state through Equation (42). In physics, S ω is called the one–particle density matrix of the system. An example of a quasi–free state is pro-vided by the tracial state (cf. Expression (83)): Definition 7 (Tracial state).
The tracial state tr U ∈ E U is the quasi–free state with symbol S tr . = H . ␈ An important density matrix ρ ( β ) ω is that related to thermal equilibrium states, or Gibbs states ω ( β ) ∈ E where β ∈ R + is the inverse temperature. In this case, given any self–dual Hamiltonian H on ( H , A ) (Definition 2), the positive operator S ( β ) H . = 11 + e − βH satisfies Condition (41) and for any A ∈ U is the symbol of a quasi–free state ω H satisfying(43) ω ( β ) H ( A ) = tr U (cid:16) A e β h B ,H B i (cid:17) tr U (cid:16) e β h B ,H B i (cid:17) . One verify that the self–dual Hamiltonian H on ( H , A ) give rise to the density matrix ρ ( β ) ∈ U ρ ( β ) . = e β h B ,H B i tr U (cid:16) e β h B ,H B i (cid:17) . Physically, ρ ( β ) minimizes the free energy of the physical system provided H . See [BR03b] for de-tails.As already discussed, for any even size Hilbert space H with associated self–dual Hilbert space ( H , A ) , the algebras U , ∧ ∗ H and Q algebras are isomorphic. More generally, for any basis projec-tion P ∈ p ( H , A ) one can endow with an involution and a norm to ∧ ∗ H in such a way that U , G P and Q are C ∗ –algebras. See Equations (28)–(29) for notations. For the sake of simplicity, let A to be U , G P or Q . By Definition 12, for any invertible bounded operator C ∈ B ( H ) providing a13ilinear element C , on A (see (11), (22) and (32)) one can define a Gaussian state ω C ∈ E A withassociated density matrix ρ C ∈ A + ∩ A , explicitly written as ρ C, A . = e α C tr A (e α C ) with α ∈ C . Simi-larly, for M ∈ R + , the operator g A, A = M e αA is called gaussian, not necessarily normalized. C iscalled the covariance of the density matrix ρ C . Note that by the isomorphism κ P of Definition 6 onecan obtain similar fermion representations at the algebras U and G P . Additionally, note that for any P ∈ p ( H , A ) and any invertible operator C ∈ B ( H ) , the isomorphism κ P of Definition 6 relatesGaussian operators g C, U ∈ U , g C, G P ∈ G P and g C, Q ∈ Q by B C,P κ P ( g C, U ) = E C,P g C, G P = D C,P κ P ( g C, Q ) , explicitly B C,P e κ P ( h B ,C − B i ) = E C,P e h H ,C − H i = D C,P e κ P ( h R ,C − R i ) , with B C,P , C
C,P , D
C,P ∈ R + positive numbers depending on C and P . See Definition 4. In particular observe that the Gibbsstate ω ( β ) H ∈ E U given by (43) is Gaussian. One can inquire about the relation between det( · ) and tr( · ) while comparing the positive numbers B C,P , E
C,P , C
C,P . See again Definition 4 and note that ourdefinitions coincide with those given in [DNP13]. In the scope of a general setting, for any covariance matrix C ( β ) H,P depending on H ∈ B ( H ) , P ∈ p ( H , A ) and β ∈ R + (see [ABPM20, Corollary 4.8]for a concrete Definition) we can write the Determinant of C ( β ) H,P as a trace of well–defined product ofoperators defined via H ∈ B ( H ) , P ∈ p ( H , A ) and β ∈ R + , see [ABPM20, Theorem 5.1]. Let W be a finite unital C ∗ –algebra. As is usual, for the tracial state tr W ∈ E W and any state ω ∈ E W with associated density matrix ρ ω ∈ W + ∩ W , the von Neumann entropy S : E W → R is given by(44) S ( ω ) . = − tr W ( ρ ω ln ρ ω ) . Similarly, for the states ω , ω ∈ E W ,(45) § ( ω k ω ) . = tr W (cid:16) ρ ω (cid:16) ln ρ ω − ln ρ ω (cid:17)(cid:17) , if supp( ρ ω ) ≥ supp( ρ ω ) , + ∞ , otherwise , denotes the entropy of ω relative to ω . In (45), for the state ω ∈ E W , supp( ω ) denotes its sup-port defined by the smallest projection P ∈ W such that ω ( P ) = 1 . The quantum entropy power associated to the state ω ∈ E W is defined by(46) E ( ω ) . = e S ( ω ) NW ∈ R + , where N W . = | W + ∩ W | ∈ N is the number of modes of the physical system described via W .Consider now the Clifford C ∗ –algebra Q given at Subsection 2.1.3. For the state ω ∈ E Q , withassociated density matrix ρ ≡ ρ ω ∈ Q + ∩ Q , and the family of self–adjoint elements { R j } j ∈ J of Q with J given by (30), the “quantum Fisher information” is(47) J ( ω R j ) . = d d θ § (cid:16) ω k ω ( θ ) R j (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) θ =0 , where ω ( θ ) R j is such that its associated density matrix is given by(48) ρ ( θ ) R j . = e θR j ρ e − θR j . Thus the “entropy variation rate” is defined by [DPR17] (see also [HKV17])(49) J ( ω ) . = X j ∈ J J ( ω R j ) . A and B be two interacting fermion systems such that the product C ∗ –algebra A I ≡ A A ⊗ A B describes the interacting system, where A A and A B are the CAR C ∗ –algebras describing A and B , respectively. For Gaussian states ω A ∈ E A ≡ E A A ⊂ A ′ A and ω B ≡ E A B ∈ A ′ B , the tensorproduct of density matrices ρ A ⊗ ρ B equals the density matrix ρ I . Moreover, by taking into accountAppendix A, the unitary bounded operator U ∈ B ( A I ) on A I satisfying UU ∗ = defines a quantumchannel given by(50) E U ( ω I ) . = tr B ( U ∗ ( ρ A ⊗ ρ B ) U ) , with ω I ≡ ω A ⊗ ω B . Then a set of results regarding entropy variation rate of ω A , ω B and ω I aresummarized in the following Corollary:C OROLLARY A and B betwofermionsystemsandtakesameassumptionsofLemma3fortheGaussianstates ω A and ω B . Therefore:1. Take α ∈ R , J ( ω C ,αR j ) = α J ( ω C ,R j ) for any R j generator of theClifford C ∗ –algebra Q C ,for C ∈ { A , B } .2. J ( ω A ) , J ( ω B ) ∈ R +0 .3. J ( ω A ⊗ ω B ) = J ( ω A ) + J ( ω B ) .4. Let U ∈ B ( A I ) be a unitary bounded operator defining a quantum channel E U . Then, for C ∈ { A , B , I } wehave J ( ω C ) ≥ J ( E U ( ω C )) . ␊ Proof.
In order to prove the assertions we remind some pivotal properties of the relative entropy. See[Lin75] and [Weh78]. Consider the product
CAR C ∗ –algebra A I ≡ A A ⊗ A B and the states ω , ω ∈ E A I . We have (i) non–negativity : § ( ω k ω ) ≥ , (ii) monotonicity : § ( ω k ω ) ≥ § ( E U ( ω ) k E U ( ω )) for any the quantum channel given by (50) and (iii) additivity : § ( ω ⊗ ω k σ ⊗ σ ) = § ( ω k σ ) + § ( ω k σ ) for any normal states ω , σ ∈ E A A and ω , σ ∈ E A B . Thus Part 1. follows from Lemma 3,while parts 2., 3 and 4 can be shown in a similar way that is done in [KS14], where the authors takeinto account (i), (ii) and (iii) properties. End
The fermionic version of the quantum entropy power inequality is stated as follows:
Theorem 1 (Fermionic Entropy Power Inequality):
Consider A and B finite fermion systems described by the Clifford C ∗ –algebras Q A and Q B , respec-tively, with N . = | Q + A ∩ Q A | = | Q + B ∩ Q B | ∈ N . Take C . = [0 , and Gaussian states ω A ∈ E Q A and ω B ∈ E Q B satisfying assumptions of Lemma 3. Then under the beam–splitter quantum channelgiven by (55) below and λ ∈ C , the concave entropy power inequality holds: E ( ω I ) ≥ λ A E ( ω A ) + λ B E ( ω B ) with λ A . = λ, λ B . = 1 − λ. ␄ Proof.
As is usual, we proceed in similar form that Blachman [Bla65]. For this, a few of supportingdefinitions are introduced:i. Take t ∈ R +0 and consider the differentiable functions t A , t B ∈ C ( R +0 ; R +0 ) such that t A (0) = t B (0) = 0 , t A ( t ) ≈ t + O ( t ) and t B ( t ) ≈ t + O ( t ) as t → ∞ . This is physically justifiedbecause of the independence of the systems A and B . See proof of Lemma 5.15i. For C ∈ { A , B , I } , let t C the time describing the evolution of the state ρ C . By Lemma 5,Expression (74) and i. one can define the function composition t I ( t ) . = λ A t A ( t ) + λ B t B ( t ) forall t ∈ R +0 .iii. For C ∈ { A , B , I } , let E (cid:16) ω C ,t C ( t ) (cid:17) be the quantum entropy power (46), such that by the deBruijin’s identity, Lemma 4, we note that ˙ E (cid:16) ω C ,t C ( t ) (cid:17) . = dd t E (cid:16) ω C ,t C ( t ) (cid:17) = 1 N E (cid:16) ω C ,t C ( t ) (cid:17) J (cid:16) ω C ,t C ( t ) (cid:17) dd t t C , Here we assume that E (cid:16) ω C ,t C (0) (cid:17) = E ( ω C ) . We can invoke the results [DMG14, Eqs. (19)and (84)] and [KS14, Corollary III-4], which are also valid for fermion systems. Namely, forany Gaussian state ω (with associated density matrix ρ ) evolving in time by the Liouvillean (63)(that is ρ t = e t L ρ , for t ∈ R +0 ), we have the following asymptotic estimate as t → ∞ (51) E ( ω t ) = e2 t + O ( t ) . iv. For C ∈ { A , B } the times satisfy the initial value problem ˙ t C . = dd t t C ( t ) = E (cid:16) ω C ,t C ( t ) (cid:17) , ˙ t C (0) = 0 . Observe that by the Peano’s Theorem for ordinary differential equations we knowthat has at least one solution.v. To lighten notations, for C ∈ { A , B , I } let E C ( t ) ≡ E (cid:16) ω C ,t C ( t ) (cid:17) and J C ( t ) ≡ J (cid:16) ω C ,t C ( t ) (cid:17) .Thus by following iii. E C (0) equals E ( ω C ) .First of all, in the Stam inequality, Lemma 5, take α = J ( ω A ) − J ( ω A ) − + J ( ω B ) − and β = J ( ω B ) − J ( ω A ) − + J ( ω B ) − : λ A J ( ω A ) + λ B J ( ω B ) ≤ J ( ω I ) . Combining last inequality with the AM–GM inequality applied to E A ( t ) J A ( t ) and E B ( t ) J B ( t ) ,and some rearrangements we get λ A E A ( t ) J A ( t ) + λ B E B ( t ) J B ( t ) ≥ ( λ A E A ( t ) + λ B E B ( t )) J A ( t ) J B ( t ) λ B J A ( t ) + λ A J B ( t ) ≥ ( λ A E A ( t ) + λ B E B ( t )) J I ( t ) . (52)With the previous notations we can study the behavior of the positive valued real differentiable func-tion f A , B , I ( t ) . = λ A E A ( t ) + λ B E B ( t ) E I ( t ) . From ii. iii. iv., v. and Inequality (52) note that f ′ A , B , I ( t ) ≥ : f ′ A , B , I ( t ) = (cid:16) λ A ˙ E A ( t ) + λ B ˙ E B ( t ) (cid:17) E I ( t ) − ( λ A E A ( t ) + λ B E B ( t )) ˙ E I ( t ) E I ( t ) = (cid:16) λ A E A ( t ) J A ( t ) ˙ t A + λ B E B ( t ) J B ( t ) ˙ t B (cid:17) − ( λ A E A ( t ) + λ B E B ( t )) J I ( t ) ˙ t I N E I ( t )= ( λ A E A ( t ) J A ( t ) + λ B E B ( t ) J B ( t )) − ( λ A E A ( t ) + λ B E B ( t )) J I ( t ) N E I ( t ) . Hence, f A , B , I ( t ) is an increasing function. Thus f A , B , I ( t → ∞ ) ≥ f A , B , I (0) , which according to i., ii.,iii. and v. yields us to ≥ λ A E A (0) + λ B E B (0) E I (0) = λ A E ( ω A ) + λ B E ( ω B ) E ( ω I ) , concluding the proof of the Theorem. End Technical results
Consider the self–dual Hilbert space ( H , A ) , and let U ′ be the self–dual CAR algebra generated bythe unit and the family { C( ϕ ) } ϕ ∈ H of elements satisfying: C ( ϕ ) ∗ is (complex) linear, C( ϕ ) ∗ . =C( A ( ϕ )) for any ϕ ∈ H and satisfies the CAR , (6). For the self–dual C ∗ –algebra U , with generators and { B( ϕ ) } ϕ ∈ H ∈ U , we consider the product C ∗ –algebra V ≡ U ⊗ U ′ such that for any ϕ , ϕ ∈ H , B( ϕ ) ⊗ ≡ B( ϕ ) ∈ V , ⊗ C( ϕ ) ≡ C( ϕ ) ∈ V so that ϕ ⊕ ϕ → B( ϕ )C( ϕ ) + C( ϕ ) B( ϕ ) = 0 . (53)Here the symbol stands for either C( ϕ ) or C( ϕ ) ∗ . Note that, due to all elements of U and U ′ are bounded, the product C ∗ –algebra V is well–defined, see [Dix77]. We define: Definition 8 (Displacement and Beam Splitter operators on self–dual
CAR –algebras).
Let ( H , A ) bea self–dual Hilbert space, and let U and U ′ be the self–dual CAR algebras as above defined. Fix abasis projection P associated with ( H , A ) and an orthonormal basis { ψ j } j ∈ J of its range h P .1. The element h C , B i on V is defined by h C , B i . = X j ∈ J C( ψ j ) ∗ B( ψ j ) , where the families of elements B | J | . = { B( ψ i ) } j ∈ J , C | J | . = { C( ψ i ) } j ∈ J will describe | J | differ-ent modes .2. Denote by (cid:16) C | J | (cid:17) T . = (cid:16) C( ψ ) , C( ψ ) ∗ , . . . , C( ψ | J | ) , C( ψ | J | ) ∗ (cid:17) and J . = ! ⊕| J | . The fermionic “Weyl displacement operator” D U (cid:16) C | J | (cid:17) : U → V , of U relative to U ′ , isdefined by D U (cid:16) C | J | (cid:17) . = e − ( C | J | , J B | J | ) U , with ( A, B ) U . = A T B for any A, B ∈ U , U ′ , and the exponential function given by (76).3. Denote by C the compact interval [0 , . The beam splitter map U λ : C → V is the unitaryoperator on B ( V ) given by U λ . = e f ( λ ) h B , C i− f ( λ ) ∗ h C , B i , fot λ ∈ C the transmissivity of the beam splitter, and f : C → C \ { } a well–defined complexfunction controlling the relative weight of the C ∗ –algebras U and U ′ , where f ( λ ) ∗ ∈ C \ { } is the conjugate complex of f ( λ ) . ␈ We can state the following [KS14]:L
EMMA D U ofDefinition8 wehavethefollowingproperties:17. Forany h B , C i and h C , B i as in Definitionwehave: D U (cid:16) C | J | (cid:17) = e h B , C i−h C , B i , and D U (cid:16) C | J | (cid:17) − = D U (cid:16) − C | J | (cid:17) = D U (cid:16) C | J | (cid:17) ∗ = D U ′ (cid:16) B | J | (cid:17) − ,
2. For any λ ∈ C , exists a complex matrix M | J | λ ∈ Mat(2 | J | , C ) such that the Heisenberg evolu-tion ofthe modes B and C isgivenby U ∗ λ D U λ = M | J | λ D , where for M λ ∈ Mat(2 , C ) , M | J | λ . = M ⊕| J | λ and D T | J | . = (cid:16) B( ψ ) , C( ψ ) , . . . , B( ψ | J | ) , C( ψ | J | ) (cid:17) . ␄ Proof.
Part 1. of Lemma is straightforward from Definition 8. For the part 2. it is enough considerthe modes B( ψ ) and C( ψ ) such that we can use Expression (77) to get U ∗ λ B( ψ )C( ψ ) ! U λ = M λ B( ψ )C( ψ ) ! , where M λ ∈ Mat(2 , C ) is a × complex matrix with components ( M λ ) , = ( M λ ) , = cos ( | f ( λ ) | ) and ( M λ ) , = sin ( | f ( λ ) | ) vuut f ( λ ) f ( λ ) ∗ and ( M λ ) , = − sin ( | f ( λ ) | ) vuut f ( λ ) ∗ f ( λ ) . Considering the matrix M ⊕| J | λ ∈ Mat(2 | J | , C ) , the proof follows. End
We interprete Part 2. of last Lemma saying that if U and U ′ describe two different fermionic sys-tems interacting between them, then the beam splitter operator U λ is used to obtain a family of out-put modes of size | J | . For technical purposes, and w.l.o.g., one usually take ran (cos ( | f ( λ ) | )) =ran (sin ( | f ( λ ) | )) = C , so that for any λ ∈ C , cos ( | f ( λ ) | ) = √ λ and sin ( | f ( λ ) | ) = √ − λ , with f ( λ ) = e i θ q | f ( λ ) | for some θ ∈ R . Then, we writte M λ,θ . = √ λ √ − λ e i θ −√ − λ e − i θ √ λ ! , which is the well–known matrix implementing the beam–splitter operator. Here, for physical purposeswe will asume that θ = 0 , so that(54) M λ . = M λ, = √ λ √ − λ −√ − λ √ λ ! , By taking into account the quantum channel given by (87) for the unitary operator U λ we interpreteby(55) ρ ( λ ) I . = E U λ ( ρ A ⊗ ρ B ) = tr B ( U ∗ λ ( ρ A ⊗ ρ B ) U λ ) the density matrix associated to the output state ω I ∈ E V .In regard to the Grassmann algebra ∧ ∗ H one can introduce Weyl displacement operators simi-larly to the self–dual CAR algebra U case (Definition 8):18 efinition 9 (Weyl displacement operator at Grassmann algebras). Let ( H , A ) be a self–dual Hilbertspace:1. Fix a basis projection P ∈ p ( H , A ) and an orthonormal basis { ψ j } j ∈ J of its range h P . Given k, l ∈ N , D h ( k ) P , h ( l ) P E . = X j ∈ J (cid:16) A ψ j (cid:17) ( k ) ∧ ψ ( l ) j .
2. For l ∈ N , define (cid:16) ψ ( l ) | J | (cid:17) T . = (cid:18) ψ ( l )1 , ( A ψ ) ( l ) , . . . , ψ ( l ) | J | , (cid:16) A ψ | J | (cid:17) ( l ) (cid:19) and J . = ! ⊕| J | . For any k ∈ N and ψ ( k ) | J | , the fermionic “Weyl displacement operator” T k : ∧ ∗ h ( k ) P → ∧ ∗ ( h ( k ) P ⊕ h ∗ ( l ) P ) , of the Hilbert space h ( k ) P relative to the Hilbert space h ( l ) P , is defined by T k (cid:16) ψ ( l ) | J | (cid:17) . = e − (cid:16) ψ ( l ) | J | , J ψ ( k ) | J | (cid:17) h P , with ( A, B ) h P . = A T B for any A, B ∈ h ( k ) P , h ( l ) P , and the exponential function of Expression(76). ␈ Straightforward calculations using Definition 9–(2), for a basis projection P ∈ p ( H , A ) , show thatthe Weyl displacement operator of Definition 9–(3) can be redefined by(56) T k (cid:16) ψ ( l ) | J | (cid:17) = e D h ( k ) P , h ( l ) P E − D h ( l ) P , h ( k ) P E , so that(57) T k (cid:16) ψ ( l ) | J | (cid:17) − = T k (cid:16) − ψ ( l ) | J | (cid:17) = T k (cid:16) ψ ( l ) | J | (cid:17) ∗ = T l (cid:16) ψ ( k ) | J | (cid:17) − , see Lemma 1 for a comparation with the Weyl displacement operator D U in the context of self–dual CAR algebras.For a fix basis projection, consider the Grassmann C ∗ –algebra G P . For the displacement operator T of Definition 9 we have:L EMMA P ∈ p ( H , A ) with range h P , and take same notationsof Definition 9. For thedisplacementoperator T wehave: T k (cid:16) ψ ( l ) | J | (cid:17) ∗ ◦ ( k ) P (cid:16) ψ ( k ) i (cid:17) ◦ ( k ) P T k (cid:16) ψ ( l ) | J | (cid:17) = (cid:16) ψ ( k ) i (cid:17) + (cid:16) ψ ( l ) i (cid:17) , foreach i ∈ J ,where ◦ ( k ) P isthecircleproductofDefinition5actingon ∧ ∗ h ( k ) . Thesymbol standsfor either ψ ( k,l ) j or (cid:16) A ψ j (cid:17) ( k,l ) . Thusforafix Hilbertspace h ( k ) P , theelement ψ ( l ) | J | is displaced as T k (cid:16) ψ ( l ) | J | (cid:17) ∗ ◦ ( k ) P ψ ( k ) | J | ◦ ( k ) P T k (cid:16) ψ ( l ) | J | (cid:17) = ψ ( k ) | J | + ψ ( l ) | J | . Let ψ ( l ) | J | and ψ ( m ) | J | well–definedon ∧ ∗ h ( l ) P and ∧ ∗ h ( m ) P ,respectively. Wehave T k (cid:16) ψ ( l ) | J | + ψ ( m ) | J | (cid:17) = T k (cid:16) ψ ( l ) | J | (cid:17) ◦ ( k ) P T k (cid:16) ψ ( m ) | J | (cid:17) e − (cid:16) ψ ( l ) | J | , J ψ ( m ) | J | (cid:17) . ␄ emark 1. Observe that this Lemma brings us to similar results proven for fermionic coherent states[OK78, CG99, CR12]. The fundamental difference between our result and those of the mentioned works,is that here we do not require a
CAR algebra and the
Grassmann numbers , which are usually introducedto provide anticommutative properties. Instead, the circle product ◦ P on Grassmann algebras providesa natural structure that combines the CAR algebra structure, Equation (25) , and the anticommutativeproperty of Grassmann algebras, see (16) . Thus, it is only necessary having Grassmann algebras endowedwith ◦ P in order to study coherent states of fermions. ␏ Proof.
Take k, l ∈ N and consider Expressions (56)–(57). Similar to Proof of Lemma 1, we can useExpressions (77)–(78) for the vector space ∧ ∗ ( h ( k ) P ⊕ h ( l ) P ) . Here, we are displacing a fix vector ψ ( k ) | J | the quantity ψ ( l ) | J | . Hence, for ψ ( k ) | J | and ψ ( l ) i of Lemma, we take into account the circle product ◦ P ofDefinition 5, Expressions (56)–(57) and (77) in order to obtain T k (cid:16) ψ ( l ) | J | (cid:17) ∗ ◦ ( k ) P ψ ( k ) i ◦ ( k ) P T k (cid:16) ψ ( l ) | J | (cid:17) = e D h ( l ) P , h ( k ) P E − D h ( k ) P , h ( l ) P E ◦ ( k ) P ψ ( k ) i ◦ ( k ) P e D h ( k ) P , h ( l ) P E − D h ( l ) P , h ( k ) P E = ψ ( k ) i + h A ( k,l ) , ψ ( k ) i i ( k ) P + 12 (cid:20) A ( k,l ) , h A ( k,l ) , ψ ( k ) i i ( k ) P (cid:21) ( k ) P . . . , with A ( k,l ) . = X j ∈ J (cid:18)(cid:16) A ψ j (cid:17) ( l ) ψ ( k ) j − (cid:16) A ψ j (cid:17) ( k ) ψ ( l ) j (cid:19) , and [ A, B ] ( k ) P is the commutator of A, B ∈ ∧ ∗ ( h ( k ) P ⊕ h ∗ ( l ) P ) obeying the circle product CAR (25) for
A, B ∈ ∧ ∗ h ( k ) P , while it is equals to zero for A, B ∈ ∧ ∗ h ( l ) P according to (16). Observe that for anybasis projection P ∈ p ( H , A ) and integers n ∈ N and k ∈ { , . . . , n } we can define(58) κ ( k ) P . = κ ( k,k )(0 , ◦ κ P For n ∈ N we can extend the isomorphism κ P of Definition 6 for any k –copy ∧ ∗ H ( k ) of ∧ ∗ H .Thus for any ϕ , ϕ ∈ H , we have ϕ ( k )1 , ϕ ( k )2 ∈ H ( k ) satisfying (25). Then, we can use this andExpression (78) in order to calculate h A ( k,l ) , ψ ( l ) i i ( k ) P as h A ( k,l ) , ψ ( k ) i i ( k ) P = X j ∈ J (cid:26)(cid:16) A ψ j (cid:17) ( k ) ◦ ( k ) P ψ ( k ) i + ψ ( k ) i ◦ ( k ) P (cid:16) A ψ j (cid:17) ( k ) (cid:27) ψ ( l ) j . Hence we are able to combine Expressions (25)–(26), from which one has h A ( k,l ) , ψ ( k ) i i P = ψ ( l ) i , andhence by (16) and notation (17) we obtain (cid:20) A ( k,l ) , h A ( k,l ) , ψ ( k ) i i ( k ) P (cid:21) ( k ) P = − (cid:20)(cid:16) A ψ j (cid:17) ( l ) ∧ ψ ( l ) i + ψ ( l ) i ∧ (cid:16) A ψ j (cid:17) ( l ) (cid:21) ψ ( k ) i = 0 . This shows that the Lemma works for ψ ( k ) j , whereas a similar calculation shows that also works for (cid:16) A ψ j (cid:17) ( k ) .For the second part, note that for any k, l ∈ N , all the elements of the form D h ( k ) P , h ( l ) P E ∈ ∧ ∗ ( h ( k ) P ⊕ h ∗ ( l ) P ) are even, which are in a commutative subalgebra (see Expression (16) and comments around it,as well as (29)). Thus for k, l, m, n, o, p ∈ N straightforward calculations arrive us to hD h ( k ) P , h ( l ) P E , hD h ( m ) P , h ( n ) P E , D h ( o ) P , h ( p ) P Eii = 0 . T k (cid:16) ψ ( l ) | J | + ψ ( m ) | J | (cid:17) = T k (cid:16) ψ ( l ) | J | (cid:17) ◦ ( k ) P T k (cid:16) ψ ( m ) | J | (cid:17) e hD h ( l ) P , h ( k ) P E − D h ( k ) P , h ( l ) P E , D h ( m ) P , h ( k ) P E − D h ( k ) P , h ( l ) P Ei = T k (cid:16) ψ ( l ) | J | (cid:17) ◦ ( k ) P T k (cid:16) ψ ( m ) | J | (cid:17) e − (cid:16)D h ( l ) P , h ( m ) P E − D h ( m ) P , h ( l ) P E(cid:17) , which is equivalent to the desired identity. End
For J defined by (30), consider the Clifford C ∗ –algebra Q ′ ≡ ( Q ′ , + , · , ∗ , k · k Q ) with generators,the unit and the family of self–adjoint elements { S j } j ∈ J , and satisfying (31) so that dim Q ′ =dim Q = 2 dim H . The product C ∗ –algebra V ′ ≡ Q ⊗ Q ′ is such a one obeying for any j ∈ J R j ⊗ ≡ R j ∈ V ′ , ⊗ S j ≡ S j ∈ V ′ so that for any i , j ∈ J R j S i + S i R j = 0 , (59)cf. (53). Because the isomorphism between Clifford C ∗ –algebras and self–dual CAR algebras, theproduct C ∗ –algebra V ′ is well–defined as well as an exponential function on V ′ , cf. (76). For θ ∈ R ,one can displace the operator A ∈ V ′ via the ∗ –automorphism ∆ θ on V ′ defined by(60) ∆ θ ( A ) . = e θ h S,R i A e − θ h S,R i , with h S, R i . = P j ∈ J S j R j ∈ V ′ , cf. displacement operator of Definition 8 and Lemma 1. By (77), weare able to write e θ h S,R i A e − θ h S,R i = A + ∞ X n =1 θ n n n ! ad n h S,R i ( A ) where for n ∈ N , ad n h S,R i ( A ) . = [ h S, R i , [ h S, R i , [ . . . , A ]] . . . ]] is the n –fold commutator of A with h S, R i . In particular, for A ≡ R i ∈ Q , with i ∈ J e θ h S,R i R i e − θ h S,R i = cos ( θ ) R i + sin ( θ ) S i . By taking cos ( θ λ ) = √ λ and sin ( θ λ ) = √ − λ , for λ ∈ (0 , we can write(61) e θλ h S,R i R i e − θλ h S,R i = √ λR i + √ − λS i θ λ . = arctan s − λλ . I.e., by using the ∗ –automorphism (60) one arrives to a fermionic quantum version of the addition ruleat the classical phase space . See [KS14, Eqs. (4) and (6)]. Now, suppose that the density matrix ρ ∈ Q + ∩ Q satisfies the quantum diffusion equation(62) dd t ρ t = L ρ t , ρ t . = e t L ρ, for any t ∈ R +0 , where by definition ρ . = ρ . See (80) below. Here, L ∈ Q is the infinitesimalgenerator or Liouvillean of the strongly continuous semigroup { e t L } t ∈ R +0 , that we will assume to bebounded, and satisfies L = 0 . Explicitly, using (81), with V j = R j , for any A ∈ Q , L is given by(63) L A = X j ∈ J ( R j [ A, R j ] + [ R j , A ] R j ) = 2 X j ∈ J ( R j AR j − A ) = − X j ∈ J [ R j , [ R j , A ]] , L , the generator of the infinite–temperature FermiOrnstein–Uhlenbeck semigroup . For details see [CM20]. By (35), for any even element A (see (10))one notes that the Liouvillean and the fermionic number operators are related by N = − L . Then,for any A , A ∈ Q + ∩ Q even elements of Q we get:(64) h L A , A i H.S. Q = h A , L A i H.S. Q , where h· , ·i H.S. Q denotes the Hilbert–Schmidt inner product on Q given by (33). One can verify that thestrongly continuous semigroup { e t L } t ∈ R +0 with the Liouvillean L defined by (63) is trace–preservingcompletely positive [GKS76], it follows that L defines a quantum channel map. For the special caseof the Clifford algebra Q , given some density matrix ρ ∈ Q + ∩ Q defining a Gaussian state ω C , theskew–symmetric matrix covariance C ≡ C ρ is given by [Bra05](65) C i , j . = h A ψ i , Cψ j i H = i2 tr Q ( ρ [ R i , R j ]) , for i , j ∈ J , and satisfying C t C ≤ | J | , where | J | ∈ Mat( | J | , C ) denotes the identity matrix, andthe symbol t denotes the transpose matrix, see (30)–(31). See also (37) and comments around it. Inabove inequality, C t C = 1 | J | holds for pure Gaussian states, that is equivalent to say that λ j = ± for j ∈ | J | in (36), [Bra05]. As proven by Bravyi–K¨onig [BK12, Lemma 1], for an initial Gaussianstate ω with density matrix ρ satisfying the differential equation (62), the Liouvillean (63) preservesthe Gaussianity of the state ω t (associated to ρ t . = e t L ρ ) for all non–negative times t ∈ R +0 .L EMMA
NTROPY VARIATION RATE ).Consider a Gaussian state ω ∈ E Q with associated density matrix ρ ≡ ρ ω ∈ Q + ∩ Q . Suppose thatthe familyofgenerators { R j } j ∈ J of Q satisfies sup j ∈ J k R j ρ k Q ∈ R + . Then, for any j ∈ J , thequantumFisherinformationisgivenby(66) J ( ω R j ) = − tr Q ( ρ [ R j , [ R j , ln ρ ]]) , in sucha waythat theentropy variationrate, see (49), is J ( ω ) = − X j ∈ J tr Q ( ρ [ R j , [ R j , ln ρ ]]) . ␄ Proof.
Fix R j , with j ∈ J , and the assumptions of Lemma. Note that for any θ, ε ∈ R + , we are ableto write § ( ω k ω ( θ + ε ) R j ) − § ( ω k ω ( θ ) R j ) = tr Q (cid:16) ρ (cid:16) ln (cid:16) ρ ( θ ) R j (cid:17) − ln (cid:16) ρ ( θ + ε ) R j (cid:17)(cid:17)(cid:17) , where § ( · , · ) is the relative entropy between two states given by (45) and ρ ( θ ) R j is the displacement of ρ according to (48) for θ ∈ R . Since ρ ∈ Q + ∩ Q we can use the identity (84), in order to obtain ln (cid:16) ρ ( θ ) R j (cid:17) − ln (cid:16) ρ ( θ + ε ) R j (cid:17) = Z ∞ (cid:16) x + ρ ( θ + ε ) R j (cid:17) − (cid:16) ρ ( θ ) R j − ρ ( θ + ε ) R j (cid:17) (cid:16) x + ρ ( θ ) R j (cid:17) − d x. (67)Define the quantity T θ,ǫ,ρ R j . = − tr Q (cid:18)Z ∞ (cid:16) x + ρ ( θ + ε ) R j (cid:17) − (cid:16) ρ ( θ ) R j − ρ ( θ + ε ) R j (cid:17) (cid:16) x + ρ ( θ ) R j (cid:17) − d x (cid:19) , so that we desire to verify that the limit lim ε → + | T θ,ǫ,ρR j | ε is bounded. By using similar arguments that in[BR03b, Example 6.2.31] we can show that (cid:12)(cid:12)(cid:12) T θ,ǫ,ρ R j (cid:12)(cid:12)(cid:12) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) ρ ( θ + ε ) R j (cid:17) − (cid:13)(cid:13)(cid:13)(cid:13) Q (cid:13)(cid:13)(cid:13) ρ ( θ ) R j − ρ ( θ + ε ) R j (cid:13)(cid:13)(cid:13) Q , Then, Q is not necessarily finite dimensional, we only require its separability. ρ ( θ ) R j − ρ ( θ + ε ) R j = e θR j ρ e − θR j (cid:16) − e − εR j (cid:17) − e θR j (cid:16) e εR j − (cid:17) ρ e − ( θ + ε ) R j , we get lim ε → + (cid:12)(cid:12)(cid:12) T θ,ǫ,ρ R j (cid:12)(cid:12)(cid:12) ε ≤ (cid:13)(cid:13)(cid:13) ρ − (cid:13)(cid:13)(cid:13) Q k R j ρ k Q , which, by the hypothesis of the Lemma ensures the boundedness of the limit lim ε → + | T θ,ǫ,ρR j | ε . On theother hand, note that lim ε → ln (cid:16) ρ ( θ ) R j (cid:17) − ln (cid:16) ρ ( θ + ε ) R j (cid:17) ε = Z ∞ (cid:16) x + ρ ( θ ) R j (cid:17) − (cid:16) ρ ( θ ) R j R j − R j ρ ( θ ) R j (cid:17) (cid:16) x + ρ ( θ ) R j (cid:17) − d x, (69)in particular, last expression can be rewritten as Z ∞ (cid:20) R j (cid:18)(cid:16) x + ρ ( θ ) R j (cid:17) − − ( x + ) − (cid:19) − (cid:18)(cid:16) x + ρ ( θ ) R j (cid:17) − − ( x + ) − (cid:19) R j (cid:21) d x = ln( ρ ( θ ) R j ) R j − R j ln( ρ ( θ ) R j ) , (70)where we had used the identity (85). It follows combining (67), (68), (69), (70) and the Lebesgue’sdominated convergence theorem that dd ε § ( ω k ω ( θ + ε ) R j ) = tr Q (cid:16) ρ h ln (cid:16) ρ ( θ ) R j (cid:17) , R j i(cid:17) . Similarly, by taking into account (67), (68), (69), (70) and the Lebesgue’s dominated convergencetheorem one more time we obtain lim θ → Z ∞ (cid:20) R j (cid:18)(cid:16) x + ρ ( θ ) R j (cid:17) − − ( x + ρ ) − (cid:19) − (cid:18)(cid:16) x + ρ ( θ ) R j (cid:17) − − ( x + ρ ) − (cid:19) R j (cid:21) d x = R j ln ( ρ ) R j − R j ln ( ρ ) − ln ( ρ ) R j + R j ln ( ρ ) R j = − [ R j , [ R j , ln ( ρ )]] , from which we deduce for any R j that the quantum Fisher information (47) is explicitly given by J ( ω R j ) = − tr Q ( ρ [ R j , [ R j , ln( ρ )]]) , and the conclusion follows. End
Then, we are in a position to state a fermionic version of the de Bruijin identity :L EMMA E B RUIJIN ’ S I DENTITY ).Let L bethe fermionic Liouvillean (63),andlet ω ∈ E Q beaGaussianstatewithassociateddensitymatrix ρ ≡ ρ ω ∈ Q + ∩ Q satisfying the differential equation (62), and such that (cid:13)(cid:13)(cid:13) ρ − t (cid:13)(cid:13)(cid:13) Q , (cid:13)(cid:13)(cid:13) d ρ t d t (cid:13)(cid:13)(cid:13) Q ∈ R + (see footnote5). Then, forany t ∈ R +0 , thefermionic de Bruijin identityholds dd t S ( ω t ) = J ( ω t ) , where S ( ω t ) and J ( ω t ) are the the von Neumann entropy (44) and the entropy variation rate (49)ofthestate ω t (provided ρ t ), respectively. ␄ roof. For the sake of completeness for a density matrix ρ ∈ Q + ∩ Q we first prove the followingwell–known relation between the Liouvillean L and the entropy S ( ρ t ) ≡ S ( ω t ) [Spo78]:(71) dd t S ( ρ t ) = − tr Q ( L ( ρ t ) ln ρ t ) , with ρ t = e t L ρ satisfying the quantum diffusion equation (62), for all t ∈ R +0 , S ( ρ t ) . = − tr Q ( ρ t ln ρ t ) and the Liouvillean (63). In fact, consider the difference between the entropies S ( ρ t + ε ) and S ( ρ t ) ,with ε ∈ R + , that is,(72) S ( ρ t + ε ) − S ( ρ t ) = − tr Q ( ρ t + ε ln ρ t + ε ) + tr Q ( ρ t ln ρ t )= − tr Q (cid:18)(cid:16) ρ t + ε − ρ t (cid:17) ln( ρ t + ε ) + ρ t Z ∞ (cid:16) x + ρ t + ε (cid:17) − (cid:16) ρ t − ρ t + ε (cid:17) ( x + ρ t ) − d x (cid:19) , where we have used some rearrangements and we took the identity (84). Define the quantity T ρ t ,ǫ . = − tr Q (cid:18) ρ t Z ∞ (cid:16) x + ρ t + ε (cid:17) − (cid:16) ρ t − ρ t + ε (cid:17) ( x + ρ t ) − d x (cid:19) , so that we desire to verify the boundedness of the limit lim ε → + | T ρt,ǫ | ε . In fact, by using similar argumentsthat in [BR03b, Example 6.2.31] we can show that (cid:12)(cid:12)(cid:12) T ρ t ,ǫ (cid:12)(cid:12)(cid:12) ≤ k ρ − t + ε k Q k ρ t + ε − ρ t k Q , and since ρ t + ε − ρ t = (cid:16) e ε L − (cid:17) e t L ρ , we obtain lim ε → + (cid:12)(cid:12)(cid:12) T ρ t ,ǫ (cid:12)(cid:12)(cid:12) ε ≤ k ρ − t k Q k L ρ t k Q , which, by the hypothesis of the Lemma ensures that lim ε → + | T ρt,ǫ | ε is bounded. Then, by the Lebesgue’sdominated convergence theorem, dd t S ( t ) . = lim ε → + S ( ρ t + ε ) − S ( ρ t ) ε can be written as dd t S ( t ) = − tr Q (cid:18) L ( ρ t ) ln( ρ t ) − ρ t Z ∞ ( x + ρ t ) − L ( ρ t ) ( x + ρ t ) − d x (cid:19) = − tr Q ( L ( ρ t ) ln( ρ t ) − L ( ρ t ))= − tr Q ( L ( ρ t ) ln( ρ t )) , where we use the representation identity ∞ R ( x + ρ t ) − ρ t ( x + ρ t ) − d x = and the Liouvilleangiven by (63), proving (71). Now, by taking into account expression (64) we can write dd t S ( ρ t ) = − tr Q ( ρ t L (ln ( ρ t ))) . Finally, one use Lemma 4 and the Liouvillean given by Expression (63) in order to get dd t S ( ω t ) = J ( ω t ) . End
We are ready to state a
Stam inequality for fermion systems:24
EMMA
TAM I NEQUALITY ).Let A and B betwointeractingfermionsystemsofthesamesize, | A | = | B | = N ,andsupposethatboth are described by the Gaussian states ω A and ω B , respectively. For C . = [0 , take λ ∈ C anddefine λ A . = λ, λ B . = 1 − λ . Forthereal numbers α, β ∈ R wehavethe Stam inequality η J ( ω I ) ≤ α J ( ω A ) + β J ( ω B ) , where η . = √ λ A α + √ λ B β and ω ( λ ) I is the output Gaussian state associated to the density matrix(55). ␄ Proof.
In order to prove the Lemma we need some pivotal results, which are similar to that given in[KS14, DMG14]. In fact, one can proof Expressions [KS14, Eq. (46)–(47)] via the matrix (54), andobserving that the covariance matrix of the composite system I ≡ A ∪ B is [Gub06] γ . = γ A γ A , B γ B , A γ B ! , with γ t γ ≤ N , γ B , A = − γ t A , B , such that the covariance of the reduced density matrix given by (55), i.e., ρ ( λ ) I . = tr B ( U ∗ λ ( ρ A ⊗ ρ B ) U λ ) is γ ( λ ) I = λ A γ A + λ B γ B − √ λ A λ B ( γ A , B + γ B , A ) whereas the displacement vector is (see Lemma 1) D( ψ j ) = q λ A B( ψ j ) + q λ B C( ψ j ) , j ∈ J, c.f., (61). Additionally, we desire to verify the compatibility of the Liouvillean (63) and the beam–splitter operator of Definition 8, see also (60). Note that the Liouvillean (63) and the quantum channelexpressed by the density matrix (55) are related by E U λ (cid:16) e t A L ⊗ e t B L (cid:17) ( ρ A ⊗ ρ B ) = e t I L E U λ ( ρ A ⊗ ρ B ) , such that the relation between covariance matrices is(73) λ A τ t A (cid:16) γ ( λ ) A (cid:17) + λ B τ t B (cid:16) γ ( λ ) B (cid:17) = τ t I (cid:16) γ ( λ ) I (cid:17) with τ (cid:16) γ ( λ ) C (cid:17) = γ ( λ ) C , C ∈ { A , B , I } . In above expressions, for C ∈ { A , B , I } , γ C , t C ∈ R +0 and τ t C (cid:16) γ ( λ ) C (cid:17) denote the covariance matrix,the time associated to the system C and the time automorphism on the algebra of complex matrices Mat( N, C ) of size N × N , respectively. Note that τ (cid:16) γ ( λ ) C (cid:17) = γ ( λ ) C . Expression (73) physically meansthat the systems A , B and I evolve independently of each other. Thus we can take, for example, thatthe times t C ∈ C ( R +0 ; R +0 ) are continuous functions of a universal time t ∈ R +0 , such that t C (0) = 0 .By (73), we observe that γ ( λ ) I = λ A γ A + λ B γ B , and denoting by τ t C ( t ) ( γ C ) ≡ τ t ( γ C ) , we find that thetime evolution of γ C is λ A τ t ( γ A ) + λ B τ t ( γ B ) = τ t ( λ A γ A + λ B γ B ) , which means that τ t : Mat( | J | , C ) → Mat( | J | , C ) is an affine transformation , i.e., the time evolution of γ ∈ Mat( | J | , C ) it must have the form τ t ( γ ) = γ + tS , where γ, S ∈ Mat( | J | , C ) are bothskew–symmetric matrices, c.f. [KS14, Lemmata III.1–III.2]. Returning to (73) we deduce that thecompatibility of times is described by(74) t I ( t ) = λ A t A ( t ) + λ B t B ( t ) with t ∈ R +0 . What is missing to verify is the compatibility of the displacement operator and the quantum channel: E U λ (cid:16) ( ρ A ) ( αθ ) R j ⊗ ( ρ B ) ( βθ ) R j (cid:17) = ( ρ I ) ( ηθ ) R j = ( E U λ ( ρ A ⊗ ρ B )) ( ηθ ) R j for α, β, θ ∈ R , some parameter η depending of α and β , and for C ∈ { A , B , I } , ( ρ C ) ( θ ) R j is given by(48) for any self–adjoint element R j of Q , the Clifford algebra, and j ∈ J defined by (30). Similar to[DMG14] by (60), one obtain that η = √ λ A α + √ λ B β . Then, straightforward calculations combinat-ing Corollary 1 and Lemmata 3, 4 yield us to the inequality: η J ( ω I ) ≤ α J ( ω A ) + β J ( ω B ) .For complete details see [DMG14]. End Open Systems, Quantum Channels and C ∗ –algebras If the physical system S interacts with its environment E , dissipation processes can occur (e.g. heatflow, diffusion processes, etc.). To tackle these kind of problems one needs results and methodsfrom quantum dynamical semigroups . In this appendix we employ basic operator algebras in orderto provide an introductory setting in the scope of open quantum systems and quantum informationtheory. In turn, we introduce relevant definitions and expressions used throughout the entire paper. Completely positive maps
Let H and H be two finite Hilbert spaces, with dimensions dim H = m and dim H = n . The tensorial product H ⊗ H is a Hilbert space such that for each ϕ ∈ H and each ϕ ∈ H , ϕ ⊗ ϕ is a bilinear form. If { ψ ,i } i ∈ I and { ψ ,i ′ } i ′ ∈ I ′ are basis of H and H respectively, then { ψ ,i ⊗ ψ ,i ′ } ( i,i ′ ) ∈ I × I ′ is a basis of H ⊗ H . In particular, the Hilbert space H ⊗ H has dimension | I | × | I ′ | = mn , its inner product is given by h ζ ⊗ ζ , ϕ ⊗ ϕ i H ⊗ H = h ζ , ϕ i H h ζ , ϕ i H , ζ , ϕ ∈ H , ζ , ϕ ∈ H , and for any A ∈ B ( H ) , B ∈ B ( H ) , A ⊗ B ∈ B ( H ⊗ H ) , such that for any ϕ ∈ H and ϕ ∈ H is satisfied ( A ⊗ B )( ϕ ⊗ ϕ ) = Aϕ ⊗ Bϕ . Note that the Hilbert spaces H and H are isomorphic to the vectorial spaces C m and C n respec-tively, whereas H ⊗ H is isomorphic to C mn . As is usual, for N ∈ N , B ( C N ) denotes the ∗ –algebraof all the continuous linear transformations of C N to C N . Hence, if N = mn , B ( C m ) and B ( C n ) will be defined as sub ∗ –algebras given by A ⊗ Mat( n, C ) and Mat( m, C ) ⊗ A . Here, for any N ∈ N , Mat( N, C ) denotes the identity map on the set of complex matrices Mat( N, C ) of size N × N . Withthis notation we are able to define: Definition 10 (Completely positive maps).
Let m, n ∈ N be two positive natural numbers. We saythat the map Φ : B ( C m ) → B ( C n ) is “positive” or “positivity preserving operator” if A ≥ impliesthat Φ( A ) ≥ . Additionally, if for any n ∈ N Φ ⊗ Mat( n, C ) : B ( C m ) ⊗ B ( C n ) → B ( C m ) ⊗ B ( C n ) . is a positivity preserving operator, we say that Φ is completely positive . ␈ An important example of completely positive maps are partial traces, which are canonically intro-duced as follows:The partial trace over B ( C n ) is the unique linear transformation Tr C n : B ( C mn ) → B ( C m ) satisfy-ing Tr C mn (( A ⊗ Mat( n, C ) ) A ) = Tr C m ( A Tr C n ( A )) . Similarly, the partial trace over B ( C m ) is the unique linear transformation Tr C m : B ( C mn ) → B ( C n ) such that Tr C mn ( A ( Mat( m, C ) ⊗ A )) = Tr C n (Tr C m ( A ) A ) . We say that an algebra W ≡ ( W , + , · , ∗ , k · k W ) is a C ∗ –algebra if it is equipped with an involution ∗ , it is complete, and it is endowed with a norm k · k W satisfying k A ∗ A k W = k A k W for all A ∈ W . W is said “unital” if it is embedded with a unit or identity operator, . If W and W are two finite C ∗ –algebras isomorphic to B ( C m ) and B ( C n ) respectively, then the product C ∗ –algebra W ≡ W ⊗ W is isomorphic to B ( C N ) , for N = mn . Here, an operator A ∈ W is view as an operator A ⊗ Mat( n, C ) ∈ W , while, the operator A ∈ W is recognized as the operator Mat( m, C ) ⊗ A ∈ W . An operator A ∈ B ( C m ) is positive if it is self–adjoint and spec( A ) ≥ . uantum dynamical semigroups Consider a unital C ∗ –algebra W , and we denote its norm by k · k W , we define: Definition 11 (Quantum dynamical semigroup).
Let W ⊂ W be a unital subalgebra of W . A semi-group on W is understood as a family P . = { P t } t ∈ R +0 ∈ B ( W ) such that for any s, t ∈ R +0 wehave(75) P s P t = P s + t , and P = . P is a “quantum dynamical semigroup (QDS)” or a “quantum Markov semigroup” if P is a stronglycontinuous semigroup (or C –semigroup), i.e., if it is continuous in the strong operator topology, thatis, for any A ∈ W one has lim t → k P t A − A k = 0 . ␈ One can check that for each t ∈ R +0 exists constants C ∈ R and D ≥ such that [EBN + k P t k W ≤ D e Ct . Note that if C = 0 , then P is bounded. Additionally, if C = 0 and D = 1 then P is said “con-tractive” or is called a “semigroup of contractions”. The semigroup is said to be an isometry if forany A ∈ W and t ∈ R +0 , we have k A P t k W = k A k W . Finally, if for any N ∈ N and t ∈ R +0 , P t ⊗ Mat( N, C ) is a positivity preserving operator, we say that P is completely positive . See Defini-tion 10. In the latter case, for any fix t in compact intervals, P t is named a quantum dynamical map.We naturally introduce the exponential function on W by(76) e A . = + ∞ X n =1 A n n ! , A ∈ W , which is an absolutely convergent series on W if and only if there is C A ∈ R +0 , such that for any n ∈ N , k A k W ≤ C nA . Note that for any bounded operators
A, B ∈ W , we are able to apply the operator expansion theorem (77) e B A e − B = A + [ B, A ] + 12! [ B, [ B, A ]] + · · · ≡ A + ∞ X n =1 ad nB ( A ) n ! , where for n ∈ N , ad nB ( A ) . = [ B, [ B, [ . . . , A ]] . . . ]] ∈ W is the n –fold commutator of A with B .Furthermore, for A, B, C, D ∈ W the identity(78) [ AB, CD ] = A { B, C } D − { A, C } BD + CA { B, D } − C { A, D } B, holds. Here, [ A, B ] . = AB − BC ∈ W , { A, B } . = AB + BA ∈ W are the usual “commutator” and“anticommutator” operators on W of A with B , respectively. Note that the series expressed in (77) isabsolutely convergent if there is D A,B ∈ R +0 , such that for any n ∈ N , k ad nB ( A ) k W ≤ D nA,B . Note that for C ∗ –algebras, one can invoke the Campbell–Baker–Hausdorff Theorem . In fact, it is astandard procedure shows that for
A, B ∈ W such that [ A, [ A, B ]] = [ B, [ A, B ]] = 0 , we have(79) e A + B = e A e B e − [ A,B ] . P t we are able to write it in terms of itsinfinitesimal generator L ∈ W (a posibly unbounded operator on W ) or Liouvillean as P t = e t L such that for any A ∈ W and t ∈ R +0 we have(80) dd t A t = L A t , A t . = e t L A. If W is a finite C ∗ –algebra, the (bounded) Liouvillean is explicitly given by the Lindblad form , thatis, [AL07](81) L A = i[ H, A ] + X i ∈ I V ∗ i [ A, V i ] + [ V ∗ i , A ] V i , where I is an index set, H ∈ W is a self–adjoint operator known as the Hamiltonian of the openquantum system described by W , and V i ∈ W . For unbounded L , there is not such explicit form forthis, see [AL07] for further details. States
Consider a unital separable C ∗ –algebra W . A linear functional ω ∈ W ∗ is a “state” if it is positive andnormalized, i.e., if for all A ∈ W , ω ( A ∗ A ) ≥ and ω ( ) = 1 . In the sequel, E W ⊂ W ∗ will denotethe set of all states on W . Note that any ω ∈ E W is Hermitian , i.e., for all A ∈ W , ω ( A ∗ ) = ω ( A ) . ω ∈ E W is said to be “faithful” if A = 0 whenever A ≥ and ω ( A ) = 0 . The set of all elements A ∈ W such that spec( A ) > will we named the positive elements of W and it will denotedby W + . Note that by the Banach–Alaoglu Theorem , E W is a compact set in the weak ∗ –topology σ ( W ∗ , W ) . Moreover, since W is unital, under the topology σ ( W ∗ , W ) , the set E W is a convex set,and its extremal points coincide with the pure states [BR03a, Theorem 2.3.15]. The latter, combiningwith the fact that W is separable allows to claim that the set of states E W is metrizable in σ ( W ∗ , W ) [Rud91, Theorem 3.16]. Note that the existence of extremal points is a consequence of the Krein–Milman Theorem . More specifically, if E ( E W ) denotes the set of extremal points of E W , E W = cch ( E ( E W )) , where, for X a Topological Vector Space and A ⊂ X , cch( A ) refers to the closed convex hull of A .Such extremal points E ( E W ) or pure states can not be written as a linear combination of any states.As an application of the extremal states is that these are used to write any “mixed state” ω ∈ E W . Bya mixed state ω ∈ E W , we mean that, there are states { ω j } mj =1 ∈ E ( E W ) , m ∈ N , and positive realnumbers, ≤ λ j ≤ for j ∈ { , . . . , m } , with m P j =1 λ j = 1 satisfying(82) ω = m X j =1 λ j ω j . In particular, if the state ω ∈ E W is pure, ω = m P j =1 λ j ω j implies that ω = ω = · · · = ω m , and λ = · · · = λ j = m . More generally, by the Choquet’s Theorem , the extremal points of E W forma Baire set G δ , and for any ω ∈ E W there is a probability measure µ ∈ M + , ( E W ) supported by E ( E W ) with barycenter ω (i.e., for all f ∈ A ( E W ) , ω ( f ) = f ( ω ) ) and satisfying µ ( E ( E W )) = 1 ,[Phe01, Isr79]. Concretely, for all f ∈ A ( E W ) and any ω ∈ E W , there is µ ∈ M + , ( E W ) such that f ( ω ) = Z E ( E W ) f ( ω ′ )d µ ω ( ω ′ ) . If C R ( E W ) is the set of all real continuous functions on E W , then A ( E W ) . = { f ∈ C R ( E W ); f ( λω +((1 − λ ) ω )) = λf ( ω ) + (1 − λ ) f ( ω ) , with λ ∈ [0 , } denotes of all the real continuous affine functions on E W .
28n this paper we will deal with faithful–normal states. By normal we mean that these are determinedby density matrices, i.e., operators ρ ∈ W + ∩ W that satisfy tr W ( ρ ) = 1 , where tr W ∈ E W is theso–called “tracial state”, which is the “normalized trace” on W , i.e., tr W ( A ) . = Tr W ( A ) / tr W ( ) , sothat Tr W ∈ W ∗ is the “trace” on W , satisfying Tr W ( AA ∗ ) = Tr W ( A ∗ A ) , for all A ∈ W . Moreprecisely, for any normal state ω ∈ E W , there exists a unique positive operator ρ ω ∈ W + ∩ W , with tr W ( ρ ω ) = 1 , such that the expectation value of A ∈ W w.r.t. ρ ω is(83) h A i ρ . = ω ( A ) = tr W ( ρ ω A ) . As a trivial case, note that tracial state tr W ∈ E W is faithful and normal. Note that for any densitymatrices ρ, ρ , ρ ∈ W + ∩ W we have the following two functional calculus representation identities ln ( ρ ) − ln ( ρ ) = Z ∞ (cid:16) ( x + ρ ) − − ( x + ρ ) − (cid:17) d x = Z ∞ ( x + ρ ) − ( ρ − ρ ) ( x + ρ ) − d x (84)and(85) ln ( ρ ) = Z ∞ (cid:16) ( x + ) − − ( x + ρ ) − (cid:17) d x. Let now { w i } i ∈ I and be the generators of the C ∗ –algebra. That is, any element A ∈ W can bewritten by(86) A = X n ∈ N X j ,...,j n ∈{ + , −} X i ,...,i n ∈ I v A ( i , . . . , i n ) w j i · · · w j n i n , where w − i . = w i and w + i . = w ∗ i , for i ∈ I , and v A ( i , . . . , i n ) : I n → C , is an I n bounded complexfunction depending on A . Note that A has not necessarily a unique form to be written. Definition 12 (Gaussian states on C ∗ –algebras). In (86) take n = 2 and A ∈ W invertible, i.e., thereis a unique element A − ∈ W such that AA − = A − A = . We say that the state ω A ∈ E W is aGaussian state associated to A if and only if its density matrix ρ A ∈ W + ∩ W can be uniquely writtenas ρ A . = e αA tr W (e αA ) , α ∈ C . For M ∈ R + , the operator g A . = M e αA ∈ W + ∩ W is called a “Gaussianoperator” associated to A . The set of all Gaussian states associated to A will be denoted by E W ,A ,whereas W A will denote the set of all Gaussians operators. ␈ Channels: Completely positive and trace preserving maps
If a physical system A is interacting (or coupled) with another one B , we usually assume that theseare described by C ∗ –algebras, namely, W A and W B , respectively. Therefore, the interacting system I ≡ A ∪ B is described by the product C ∗ –algebra W I ≡ W A ⊗ W A . Then, for any states ω A ∈ E W A and ω B ∈ E W B the interacting state ω I ∈ E W I is explicitly given by ω I = ω A ⊗ ω B . If the states arenormal states, it follows that their associated density matrices are given by ρ I = ρ A ⊗ ρ B , so that ω I ( A ) = tr W I (( ρ A ⊗ ρ B ) A ) , A ∈ W I . Let F U : W → W be the ∗ –automorphism on W defined uniquely by F U ( A ) . = U ∗ A U , A ∈ W . Here, U ∈ B ( W ) is a unitary bounded operator on W , which implemented the ∗ –automorphism F U .Thus if I ≡ A ∪ B is the interacting system mentioned above, then the operation E U : W A ⊗ W B → W A defined by(87) E U ( ω I ) . = tr B ( F U ( ρ I )) ≡ tr B ( U ∗ ( ρ A ⊗ ρ B ) U ) reduced system A [NC10], where tr B is the normalized partial trace over W B . In thispaper we will say that the ∗ –automorphism F U is “trace preserving” if it is valid the condition(88) UU ∗ = , In particular, we say that the map E U is a “quantum channel” with inputs A , B and output C if itis completely positive and is trace preserving (CPTP). Recall that the partial trace is a completelypositive map, and hence E U is a well–defined quantum channel. Acknowledgments:
This work is supported by
Departamento de F´ısica of Universidad de Los Andes .We are very grateful to A. Vershynina and N. Datta for hints and discussions.
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